Bi-Hamiltonian structures of KdV type**To Franco Magri on the occasion of his 70th birthday, with friendship and admiration.

Abstract

Combining an old idea of Olver and Rosenau with the classification of second and third order homogeneous Hamiltonian operators we classify compatible trios of two-component homogeneous Hamiltonian operators. The trios yield pairs of compatible bi-Hamiltonian operators whose structure is a direct generalization of the bi-Hamiltonian pair of the KdV equation. The bi-Hamiltonian pairs give rise to multi-parametric families of bi-Hamiltonian systems. We recover known examples and we find apparently new integrable systems whose central invariants are non-zero; this shows that new examples are not Miura-trivial.